hh-pp spectral element methods for three dimensional elliptic problems on non-smooth domains, Part-III: Error estimates, preconditioners, computational techniques and numerical results

The present paper is the third of a series of papers devoted to the study of hh-pp spectral element methods for three dimensional elliptic problems on non-smooth domains using parallel computers. In this paper we provide error estimates, preconditioners and numerical results. The spectral element functions are fully non-conforming. We propose preconditioners on non-smooth domains which can be diagonalized using separation of variables technique. Optimal error estimates in terms of number of layers in the geometrical mesh and in terms of number of degrees of freedom are obtained. The method is easy to implement on a parallel computer and we briefly outline computational techniques. We give results of numerical simulations to confirm the theoretical estimates. Theoretical results have been also validated by computational experiments which are published independently in Dutt et al. (2014).